Proceedings of the Royal Society of Edinburgh: Section A Mathematics

Research Article

The critical exponents of parabolic equations and blow-up in Rn

Yuan-wei Qia1

a1 Department of Mathematics, Hong Kong University of Science and Technology, Hong Kong

In this paper we study the Cauchy problem in Rn of general parabolic equations which take the form ut = Δum + ts|x|σup with non-negative initial value. Here s ≧ 0, m > (n − 2)+/n, p > max (1, m) and σ > − 1 if n = 1 or σ > − 2 if n ≧ 2. We prove, among other things, that for ppc, where pcm + s(m − 1) + (2 + 2s + σ)/n > 1, every nontrivial solution blows up in finite time. But for p > pc a positive global solution exists.

(Received October 10 1996)

(Revised January 23 1997)